Hi all,

I have an experimental data set of \([(x_1,y_1) (x_2,y_2) ... (x_n,y_n)]\). Where (now) \(x_i's\) are the predictor variables (independent, no uncertainty) and the \(y_i's\) are the response variables (dependent). I know the \(y_i's\) to have a covariance matrix \(\Sigma\). Or in other words, I have a vector of variances, \(\sigma^2\), associated with each of the \(y_i\) measurements and I believe that the \(y_i's\) are correlated by a known correlation matrix, say, \(\rho\), due to my measurement technique. My goal is fit a regression line through the data and make some statistical inferences from the line of best fit. The inferences require that the error terms, \(e_i\), be distributed like \(N(0,\Sigma')\) (\(\Sigma'\) not necessarily equal \(\Sigma\)).

As far as I understand, I cannot use something like Pearson's chi-squared test because that assumes that the \( x_i's \) are independent. Mine are not.

So, I came to the conclusion that I must be testing against a multivariate distribution rather than a univariate distribution. However, if we are speaking of a multivariate distribution, my sample size is only 1. So, the way I understand it, that would be like asking if a single experimental measurement came from a univariate normal distribution. Am I right about this? How should this correlated data be analyzed?

If my data pass a univariate normality test, can I carry on with the regression analysis using multivariate techniques?

Thanks!

mity

I have an experimental data set of \([(x_1,y_1) (x_2,y_2) ... (x_n,y_n)]\). Where (now) \(x_i's\) are the predictor variables (independent, no uncertainty) and the \(y_i's\) are the response variables (dependent). I know the \(y_i's\) to have a covariance matrix \(\Sigma\). Or in other words, I have a vector of variances, \(\sigma^2\), associated with each of the \(y_i\) measurements and I believe that the \(y_i's\) are correlated by a known correlation matrix, say, \(\rho\), due to my measurement technique. My goal is fit a regression line through the data and make some statistical inferences from the line of best fit. The inferences require that the error terms, \(e_i\), be distributed like \(N(0,\Sigma')\) (\(\Sigma'\) not necessarily equal \(\Sigma\)).

As far as I understand, I cannot use something like Pearson's chi-squared test because that assumes that the \( x_i's \) are independent. Mine are not.

So, I came to the conclusion that I must be testing against a multivariate distribution rather than a univariate distribution. However, if we are speaking of a multivariate distribution, my sample size is only 1. So, the way I understand it, that would be like asking if a single experimental measurement came from a univariate normal distribution. Am I right about this? How should this correlated data be analyzed?

If my data pass a univariate normality test, can I carry on with the regression analysis using multivariate techniques?

Thanks!

mity

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